Both words mean to talk over something. The first difference between them is that argue can only be a verb form, while debate can either be a verb or a noun. Debate means, as a noun, strife or discord. It also refers to an informal but generally civil discussion of opposing points of views. As a verb, it can mean to fight, to dispute, or argue, especially in a public area. As it was already mentioned before, argue in this form can only be a verb. It means to prove something, to indicate or to imply. It also means to have an argument or a quarrel about something, or to present a point of view.
